Diversification Penalties
An unexpected Stochastic Dominance
Overview
Main Idea of the paper
Stochastic Dominance
Pareto Distributions and fat tails
Real world examples of fat tails
Conclusions of the paper
Considerations for our work
Main Idea
Specific - There are instances where a concentration of risk is preferable to a diversified portfolio. For example, when losses are Pareto distributed.
General - We make assumptions, shortcuts, heuristics all the time. That’s fine, but we need to be aware when our assumptions break down.
Stochastic Dominance
\[
X \le_{st} Y \leftrightarrow P(X \le x) \ge P(Y \le x) \forall x \in \mathbb{R}
\]
A partial order of random variables
“\(X\) is smaller than \(Y\) in terms of first-order stochastic dominance”
Stochastic Dominance
Stochastic Dominance
The relation \(X \le_{st} Y\) means that all decision makers with an increasing utility function will prefer the loss \(X\) to the loss \(Y\)
(page 2)
Pareto Distribution
\[
CDF: P_{\alpha,\theta} = 1 - \left(\frac{\theta}{x}\right)^\alpha, x \ge \theta\\
Mean = \begin{cases}
\infty & \alpha\le 1 \\
\frac{\alpha\theta}{\alpha - 1} & \alpha > 1
\end{cases}\\
\]
\[
\alpha \leftarrow \text{$``$shape''}\\
\theta \leftarrow \text{$``$scale''}
\]
Pareto Distribution
Infinite mean on the distribution itself, not on (a finite set of) samples from the distribution.
Pareto Distribution
Examples of Pareto Risks
Earthquakes \(\alpha \in [0.5, 1.5]\)
Wind (some) \(\alpha \approx 0.7\)
Nuclear power accidents \(\alpha \in [0.6, 0.7]\)
Operational losses in banking \(\alpha \in [0.7, 1.2]\)
Cyber losses \(\alpha \in [0.6, 0.7]\)
Swiss Solvency Test (airline losses) \(\alpha = 1\)
Commercial Property losses at 2 Lloyds Syndicates \(\alpha\) “considerably less than 1”
Number of deaths in earthquakes and pandemics
(page 5)
Key Conclusions of the Paper
For \(X, X_1, ..., X_n \sim \text{Pareto}(\alpha), \alpha \in (0,1]\)
For \(\theta_1, ..., \theta_n \in \Delta_n\),
\[
X \le_{st}\sum_{i = 1}^n \theta_i X_i
\]
And for \(t > 1\),
\[
P\left(\sum_{i = 1}^n \theta_i X_i > t \right) > P(X > t)
\]
if \(\theta_i > 0\) for at least 2 \(i \in [n]\)
(Theorem 1, page 6)
Key Conclusions of the Paper
Having shares in multiple Pareto risks is less preferable to having full exposure to a single Pareto Risk
(This is not the case if the underlying distributions have finite mean)
Key Conclusions of the Paper
Generalised to random weights of random counts of Pareto Risks
It is less risky to insure one large policy than to insure any independent policies of the same type of ultra-heavy-tailed Pareto losses and thus the basic principle of insurance does not apply to ultra-heavy-tailed Pareto losses
(page 9)
Key Conclusions of the Paper
Also shows that under pooling of risk (e.g. sharing CAT losses):
Each insurer expects to suffer lower losses on their loss,
but each insurer will have a higher frequency of bearing a loss.
The combination leads to a higher probability of default of the insurer at any capital reserve level
(page 12)
Key Conclusions of the Paper
An insurance market cannot exist among a set of insurers each exposed to their own Pareto loss.
But under certain circumstances, including external entities with sufficient risk appetite can produce a market. Reinsurance still works.
My Conclusions
Diversification Penalties can exist
Heavy tailed distributions are unintuitive, but more common than we might think
We make many assumptions and simplifications, generally fine, but sometimes not
You don’t really understand your tools if you don’t know when not to use them